| Exam Board | OCR MEI |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Marks | 18 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Implicit equations and differentiation |
| Type | Show dy/dx equals given expression |
| Difficulty | Standard +0.3 This is a straightforward multi-part question testing standard C4 techniques: verifying parametric equations (substitution), parametric differentiation (chain rule), sketching scaled curves, perpendicular gradients, and solving a separable differential equation. All parts follow routine procedures with no novel insight required, making it slightly easier than average. |
| Spec | 1.03g Parametric equations: of curves and conversion to cartesian1.07s Parametric and implicit differentiation1.08k Separable differential equations: dy/dx = f(x)g(y) |
A curve has equation
$$x^2 + 4y^2 = k^2,$$
where $k$ is a positive constant.
\begin{enumerate}[label=(\roman*)]
\item Verify that
$$x = k\cos\theta, \quad y = \frac{k}{2}\sin\theta,$$
are parametric equations for the curve. [3]
\item Hence or otherwise show that $\frac{dy}{dx} = -\frac{x}{4y}$. [3]
\item Fig. 8 illustrates the curve for a particular value of $k$. Write down this value of $k$. [1]
\end{enumerate}
\includegraphics{figure_8}
\begin{enumerate}[label=(\roman*)]
\setcounter{enumi}{3}
\item Copy Fig. 8 and on the same axes sketch the curves for $k = 1$, $k = 3$ and $k = 4$. [3]
\end{enumerate}
On a map, the curves represent the contours of a mountain. A stream flows down the mountain. Its path on the map is always at right angles to the contour it is crossing.
\begin{enumerate}[label=(\roman*)]
\setcounter{enumi}{4}
\item Explain why the path of the stream is modelled by the differential equation
$$\frac{dy}{dx} = \frac{4y}{x}.$$ [2]
\item Solve this differential equation.
Given that the path of the stream passes through the point (2, 1), show that its equation is $y = \frac{x^4}{16}$. [6]
\end{enumerate}
\hfill \mbox{\textit{OCR MEI C4 Q4 [18]}}